Problem: $\dfrac{dy}{dx}=y(2x-5)$ Is $y=-e^{x^2-5x}$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Answer: In order to find whether $y=-e^{x^2-5x}$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $y$, we need to find the corresponding $\dfrac{dy}{dx}$ expression to substitute into the equation: $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{d}{dx}\left[-e^{x^2-5x}\right] \\\\ &=-(2x-5)e^{x^2-5x} \end{aligned}$ Now we substitute ${y=-e^{x^2-5x}}$ and ${\dfrac{dy}{dx}=-(2x-5)e^{x^2-5x}}$ into the equation: $\begin{aligned} {\dfrac{dy}{dx}}&={y}(2x-5) \\\\ {-(2x-5)e^{x^2-5x}}&\stackrel{?}{=}\left({-e^{x^2-5x}}\right)(2x-5) \\\\ -(2x-5)e^{x^2-5x}&\stackrel{\checkmark}{=}-(2x-5)e^{x^2-5x} \end{aligned}$ We obtained the same expression on each side. In conclusion, yes, $y=-e^{x^2-5x}$ is a solution to the differential equation.